Consider the function \(\mathrm f(x)=\dfrac{ax}{x+b}\), \(x\in\mathbb R\), \(x\neq-b\).
Given that \(\mathrm f\) is a self-inverse function, find a relationship between \(a\) and \(b\).[3]
Suppose \(a=2\) and \(b=-3\).
Sketch the graph of \(y=\mathrm f(x)\).[3]
Find \(\mathrm f^{-1}(x)\) and state its domain.[3]
Given that the graph of \(y=\mathrm f(x)\) is translated by \(\begin{pmatrix}r\\s\end{pmatrix}\) to obtain the graph of \(y=\mathrm f^{-1}(x)\), state the value of \(r\) and of \(s\).[3]
The graph of \(y=\mathrm f(x)\) undergoes the following ordered transformations:[3]
A: Horizontal stretch of scale factor 2.
B: Horizontal translation by −6 units.
C: Reflection in the x-axis.
Find the expression of the resulting function.